A Koopman Operator Tutorial with Othogonal Polynomials

TL;DR

This paper provides a comprehensive tutorial on the Koopman Operator theory, demonstrating its application to solving differential equations using orthogonal polynomials and illustrating the implementation with MATLAB code for the Duffing oscillator.

Contribution

It offers a detailed explanation of Koopman theory and a practical MATLAB implementation using Legendre polynomials for solving dynamical systems.

Findings

Koopman Operator can analytically solve ODEs via eigenfunction expansion.

Legendre polynomials serve as effective orthogonal basis functions.

MATLAB code implementation successfully solves the Duffing oscillator.

Abstract

The Koopman Operator (KO) offers a promising alternative methodology to solve ordinary differential equations analytically. The solution of the dynamical system is analyzed in terms of observables, which are expressed as a linear combination of the eigenfunctions of the system. Coefficients are evaluated via the Galerkin method, using Legendre polynomials as a set of orthogonal basis functions. This tutorial provides a detailed analysis of the Koopman theory, followed by a rigorous explanation of the KO implementation in a computer environment, where a line-by-line description of a MATLAB code solves the Duffing oscillator application.